# Surface specific asperity model for prediction of friction in boundary and mixed regimes of lubrication

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## Abstract

Machine downsizing, increased loading and better sealing performance have progressively led to thinner lubricant films and an increased chance of direct surface interaction. Consequently, mixed and boundary regimes of lubrication are prevalent with ubiquitous asperity interactions, leading to increased parasitic losses and poor energy inefficiency. Surface topography has become an important consideration as it influences the prevailing regime of lubrication. As a result a plethora of machining processes and surface finishing techniques have emerged. The stochastic nature of the resulting topography determines the separation at which asperity interactions are initiated and ultimately affect the conjunctional load carrying capacity and operational efficiency. The paper presents a procedure for modelling of asperity interactions of real rough surfaces, from measured data, which do not conform to the usually assumed Gaussian distributions. The model is validated experimentally using a bench top reciprocating sliding test rig. The method demonstrates accurate determination of the onset of mixed regime of lubrication. In this manner, realistic predictions are made for load carrying and frictional performance in real applications where commonly used Gaussian distributions can lead to anomalous predictions.

## Keywords

Real rough surfaces Contact load carrying capacity Friction## List of symbols

- \( {\tilde{\text{A}}} \)
Mean area of asperity contact in the apparent area of contact

- \( {\mathcal{A}} \)
Apparent area of contact

*a*Acceleration of floating plate

*b*Strip face-width

*c*Lubricant rupture boundary

*d*Surface separation of mean centrelines

- \( d_{i} \)
Height above the centreline of a surface

- \( E^{\prime } \)
Composite modulus of elasticity

*f*Total friction

- \( f_{b} \)
Boundary friction

- \( f_{v} \)
Viscous friction

- \( {\text{F}}_{5/2} \)
Statistical function for asperity load carrying capacity

- \( {\text{F}}_{2} \)
Statistical function for asperity contact area

*h*Separation or film thickness

- \( h_{0} \)
Initial gap between surfaces

- \( h_{ini} \)
Height of the highest peak relative to the mean centre-line of the surface

*m*Mass of floating plate

*L*Length of sliding strip

- \( p \)
Hydrodynamic pressure

- \( p_{atm} \)
Atmospheric pressure

- \( p_{cav} \)
Cavitation pressure

- \( P \)
Load carried by one asperity pair

- \( \tilde{P} \)
Mean asperity load carried for the apparent area of contact

*r*Radial distance between asperity tips

*R*Crown radius of the ring parabolic profile

*t*Time

*U*Sliding velocity

*w*Peak penetration

- \( w_{p} \)
Vertical interference between asperity tips

*W*Applied load on the strip (total contact load)

*W*_{hy}Hydrodynamic reaction

- \( {\text{z}}_{i} \)
Topography height above the centreline of surfaces

*Z*Pressure–viscosity index

## Greek symbols

*α*Piezo-viscosity

*β*Asperity tip radius

*δ*Surface deformation

*η*Lubricant dynamic viscosity

*η*_{0}Lubricant dynamic viscosity at ambient conditions

*λ*Stribeck parameter,

*λ*=*h*/*σ**ξ*Asperity density per unit area

*ρ*Lubricant density

*ρ*_{0}Lubricant density at ambient conditions

*ς*Coefficient of the boundary shear strength

*ϕ*Probability distribution function

*ϕ*^{*}Convoluted probability distribution function

*σ*Root mean square variation from mean surface centreline

*τ*_{0}Eyring shear stress of the lubricant

- Ψ
Asperity tip curvatures

## 1 Introduction

A sufficiently thick low shear strength film of lubricant is usually desired to form in all contact conjunctions to carry the applied load and guard against the direct interaction of asperities on the opposing surfaces, thus reducing frictional losses. However, in practice this ideal situation is often not achieved, owing to many circumstances, such as stop–start or reciprocating motions, which affect the entrainment of the lubricant into the contact with relative motion of surfaces [1, 2]. There may also be a lack of lubricant availability at the inlet to a conjunction as well as reverse and swirl flows there [3, 4]. As the result, many contacts suffer from a lack of a coherent lubricant film where a proportion of applied load is carried by the ubiquitous asperities on the counterface surfaces. These interactions increase the generated friction, for example in the cases of piston-cylinder system at dead centre reversals [5, 6, 7, 8] and cam-follower contact in the inlet reversal positions [9]. Various palliative actions are undertaken in order to mitigate these adverse effects within the mixed regime of lubrication, including the use of hard wear-resistant coatings. Another approach has been surface modifications such as cross-hatching of cylinder surfaces or texturing in order to entrap reservoirs of lubricant for instances of poor entraining motion. In order to opt for any method of palliation, it would be instructive to initially predict the extent of boundary interactions in an accurate manner.

Greenwood and Tripp [10] and Greenwood and Williamson [11] provided mathematical discourses for asperity interactions between pairs of rough surfaces for simplified asperity geometry and an assumed Gaussian distribution of peak heights. A further series of assumptions were made, most crucially an average asperity tip radius and an average indentation depth at given separations with the mutual approach of rough counterfaces. The method also accounted for the oblique interaction of any pair of opposing asperities. These are usually regarded as necessary assumptions to deal with the complex nature of rough surface topography. The Gaussian assumption made in Greenwood and Tripp [10] is not a prerequisite of the method. Consideration of typical rough surfaces suggests that a peak height distribution would have a mean value above the mid-line of the surface as the majority of the peaks would reside in the upper reaches of the surface. It is, therefore, advantageous to develop an asperity model which allows more representative peak height distributions to be utilised, particularly for lubricated contacts which were not discussed by Greenwood and Tripp [10].

Greenwood and Tripp [10] used a Gaussian distribution to approximate the probability of peak height interactions at a given surface separation; \( \lambda \) where this is inversely proportional to \( \sigma \) (root mean square; RMS) of the surface heights. As a result the mean of the peak height distribution is set to the same mean as the surface height distribution as a simplification for the proof of concept which, for engineering surfaces, tends to predict that the first surface interactions occur at a lower value of \( \lambda \) than is otherwise the case. This results in an estimate of asperity load carrying capacity significantly below a real case at the upper reaches of the surface and a lower separation limit at which the mixed regime of lubrication is expected to commence. One way to deal with this shortcoming is to measure the mean peak height relative to the surface centre-line and shift the mean of the Gaussian distribution to this location. This requires the analysis of the surface and identification of the peak height distribution. Therefore, “the surface specific distribution” might as well be used for the rest of the analysis, thus improving the accuracy of surface representation with little extra effort. Furthermore, the Gaussian distribution used to describe the peak height distribution must have the same standard deviation as the actual peak height distribution, not the RMS roughness, \( \sigma \), which is derived from the surface height distribution.

Greenwood and Williamson’s model [11] was further developed to account for non-uniform radii of curvature of asperity peaks by Hisakado [12], and for elliptic paraboloid asperities by Bush et al. [13], as well as for anisotropic surfaces by McCool [14]. The Greenwood and Tripp model [10] was extended by Pullen and Williamson [15] to account for the plastic deformation of asperities and further improved by Cheng et al. [16] for an elasto-plastic model. A recent extension of the model for combined elasto-plastic and adhesion of asperities for fairly smooth surfaces, using fractal geometry was reported by Chong et al. [17]. Nevertheless, the original Greenwood and Tripp model [10] has been widely used in many applications [18, 19, 20, 21].

## 2 Background Theory

The surface specific contributions are the peak height distributions, \( \phi \left( {z_{1} } \right) \) and \( \phi \left( {z_{2} } \right) \), and the asperity peak density, \( \xi \). Examples of measured surface height distributions can be seen in the work of Peklenik [22] who also discussed some of the key variations between measured and generated surface properties.

The function, \( F_{5/2} \left( \lambda \right) \), represents the statistical likelihood of interactions of deformed asperities at all separations as one surface is lowered onto the other. In order to improve the accuracy of the model for a specific pair of surfaces the specific parameter must be calculated. The parameters; \( \xi \), \( \beta \) and \( \sigma \) can be readily determined by many commercially available metrology systems. Determination of the function \( F_{5/2} \left( \lambda \right) \) from measured surface topography is more involved. As already noted, to correctly apply the assumption of a Gaussian distribution of peak heights, the standard deviation and mean of the peak heights are needed a priori. However, it is more representative to determine the peak height distributions of the surfaces through a simple extension of the approach highlighted above, using instead the surface specific data.

\( F_{5/2} \left( \lambda \right) \) and the roughness parameter determine the probable number of asperities which are penetrated at a given surface separation and the extent of their deformation.

## 3 Extension to measured surface data

Measuring surface topography using a variety of measurement techniques yields an array of surface height data at discrete measured nodes. The discrete surface data is then used in a probabilistic model, such as that described above, with the frequency distributions required in the form of a histogram of each surface. The distribution is dependent on the size and resolution of the sampled area, as well as the surface itself.

Then, the calculation of functions such as \( {\text{F}}_{5/2} \left(\uplambda \right) \) is a routine matter for any desired surface height through integration of the histogram columns residing above the \( \uplambda \) value being considered with a weighting as shown in Eq. (16).

The remaining surface specific parameters considered in this model are: \( \xi \), \( \beta \), \( \Psi \) are and \( \sigma \). The determination of \( \beta \) and \( \Psi \) not considered satisfactory techniques for measured surfaces. \( \beta \) represents the mean radius of all the asperity pairs in contact at a given separation. For multi-scale surfaces, such as cross-hatched cylinder liners, the manufacturing process often comprises several stages. Honing creates a rough, isotropic surface with a negatively skewed surface height distribution by machining away the upper levels of the surface with successively smoother machining processes. This means that there is a variation in \( \beta \) at different heights of the surface. At the upper levels there are likely to be smoother and more rounded asperities, whilst any lower peaks formed by the initial machining process and untouched by any subsequent operation would have a lower asperity radius (i.e. sharper peaks). As a result a variable \( \beta \) value is encountered at different separations.

## 4 Results for measured surfaces

It can be seen from the convolution of the peak height distributions (Fig. 5a) that there are significantly more asperities above the centre-line than would be predicted by a Gaussian distribution with a mean of 0. Not only is there a non-Gaussian peak height distribution, but also the mean height is non-zero which can also be seen for surface 2 with approximately Gaussian peak height and surface height distributions. This was given in the curve fit process of Greenwood and Tripp [10] or subsequent studies which have used this distribution assumption.

Figure 6 shows significant differences between the calculated functions. This is in part due to the increased proportion of asperities in the higher regions of the surface for the case of surface specific data. There is also a greater cumulative effect in these differences as the surfaces are moved together and the asperities continue to deform to a greater extent. For this case, the Gaussian assumption under predicts the \( F_{5/2} \) function for these surfaces by over 80 %.

Figure 8 shows a significant variation in \( \beta \left( h \right) \) at the level of the upper asperities even though surface 2 has a minimal radius variation (Fig. 7b). The characteristic equations for asperity load share (Eq. 14) and contact area (Eq. 15) show that the variation in the asperity radius with mutually converging surfaces would significantly alter the load carrying capacity and thus any predicted generated friction. To apply the variables \( \beta \left( h \right) \) and \( \Psi \left( h \right), \) sixth order polynomial curve fits are used in the numerical model. The full range of the surface height distributions has bee shown in Figs. 7 and 8 to demonstrate the convergence to the mean. On the other hand, the Greenwood and Tripp model [10] is concerned only with the upper reaches of the topography.

## 5 Validation of methology

The methodology for representation of contact of a pair of rough surfaces (described above) using surface specific data is validated against experimental measurements of friction from a precision sliding tribometer.

### 5.1 Sliding tribometer

*f*) equates the inertial force measured by the load cells as:

Lubricant data

Parameter | Value | Unit |
---|---|---|

Eyring shear stress ( | 2 | MPa |

Lubricant density ( | 839.3 (at 20 °C) | Kg/m |

Lubricant dynamic viscosity (\( \eta_{0} \)) | 0.1583 (at 20 °C) | ×10 |

The rig is operated at low sliding velocity, representative of poor kinematic conditions of piston motion near the dead centre reversals and in this case under isothermal conditions and at relatively low load. The load is applied through the loading mechanism shown in the figure. Mixed or boundary regimes of lubrication would be expected under these conditions. Further details about the slider rig and its operation are provided in Morris et al. [24].

Relatively low load and large apparent contact area, \( {\mathcal{A}} \) (contacting face of the strip) yields insufficient lubricant pressures to cause any elasto-hydrodynamic deformation of the surfaces. Furthermore, any lubricant film thickness, *h*, above the mean surface height of topography, *d*, is expected to be quite thin, yielding a mixed regime of lubrication.

### 5.2 The numerical model

The first term on the right-hand side of Eq. (28) represents the non-Newtonian shear of thin pockets of lubricant. The second term corresponds to the direct interaction of asperities. \( {\varsigma } \) is the coefficient of shear strength of asperities, measured using an atomic force microscope in the lateral force mode [8]. The Eyring shear stress is given in Table 1, and \( {\varsigma } = 0.17 \) [26].

*h*is required in order to predict the viscous friction contribution in Eq. (27). Therefore, a quasi-static load balance must be sought between the contact load carrying capacity, comprising asperity load share and any hydrodynamic lubricant reaction against the applied load, thus:

*U*is the sliding velocity of the strip in the

*x*-direction. The strip is of finite length, thus the Poiseuille side-leakage flow due to pressure gradient in the lateral

*y*-direction is also taken into account. The final term on the right-hand side of the equation takes the squeeze film effect into account due to the transient nature of the problem.

These correspond to atmospheric pressure at the inlet (\( x = - {\raise0.7ex\hbox{$b$} \!\mathord{\left/ {\vphantom {b 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \)), where *b* is the effective face-width of the sliding strip and Swift–Stieber boundary conditions at the film rupture point at the contact exit constriction (\( x = c \)), where the cavitation pressure is that of the lubricant at the environmental temperature of 20 °C. In this study the cavitation pressure is assumed to be equal to the atmospheric pressure.

A second order finite difference method is used to solve Reynolds equation by utilising Point-Successive Over-Relaxation scheme. During the iterations the lubricant properties are also updated. The procedure used, including for convergence criteria, are described in Rahmani et al. [29].

## 6 Results and discussion

*h.*Therefore, the results illustrate the importance of considering the asperity radii as a variable with respect to separation (i.e. \( \beta \left( h \right) \)). The progressive improved predictions account for small gains in accuracy, but represent significant gains in practice.

## 7 Conclusions

The current analysis highlights the potential mis-representation which may occur as a result of assuming a Gaussian surface roughness distribution in modelling asperity interactions. Considering real surface-specific roughness distributions and comparing with experimental data the potential difference is clearly demonstrated. The repercussions of this for asperity load carrying capacity and also the onset of mixed regime of lubrication is described.

It is also clear that the method of considering variation in asperity radii at different separations, developed here, offers additional improvements for surface analysis and modelling. The use of measured surface-specific distribution and separation-dependent variable asperity radii are the main contributions made to knowledge in this paper. The developed methodology also shows very good agreement with experimental measurements of friction.

## Notes

### Acknowledgments

The authors would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC) for the sponsorship of this research under the Encyclopaedic Program Grant (www.encyclopaedic.org).

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